L(s) = 1 | + (−1.07 − 0.915i)2-s + (1.49 + 0.881i)3-s + (0.324 + 1.97i)4-s + 1.83·5-s + (−0.800 − 2.31i)6-s + 4.30i·7-s + (1.45 − 2.42i)8-s + (1.44 + 2.62i)9-s + (−1.97 − 1.67i)10-s − 5.53i·11-s + (−1.25 + 3.22i)12-s − 1.97i·13-s + (3.93 − 4.63i)14-s + (2.72 + 1.61i)15-s + (−3.78 + 1.28i)16-s + 5.74i·17-s + ⋯ |
L(s) = 1 | + (−0.762 − 0.647i)2-s + (0.860 + 0.509i)3-s + (0.162 + 0.986i)4-s + 0.818·5-s + (−0.326 − 0.945i)6-s + 1.62i·7-s + (0.514 − 0.857i)8-s + (0.481 + 0.876i)9-s + (−0.624 − 0.529i)10-s − 1.66i·11-s + (−0.362 + 0.931i)12-s − 0.547i·13-s + (1.05 − 1.23i)14-s + (0.704 + 0.416i)15-s + (−0.947 + 0.320i)16-s + 1.39i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49880 + 0.379263i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49880 + 0.379263i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.07 + 0.915i)T \) |
| 3 | \( 1 + (-1.49 - 0.881i)T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 1.83T + 5T^{2} \) |
| 7 | \( 1 - 4.30iT - 7T^{2} \) |
| 11 | \( 1 + 5.53iT - 11T^{2} \) |
| 13 | \( 1 + 1.97iT - 13T^{2} \) |
| 17 | \( 1 - 5.74iT - 17T^{2} \) |
| 19 | \( 1 - 5.40T + 19T^{2} \) |
| 29 | \( 1 + 7.03T + 29T^{2} \) |
| 31 | \( 1 + 3.45iT - 31T^{2} \) |
| 37 | \( 1 - 6.42iT - 37T^{2} \) |
| 41 | \( 1 - 1.85iT - 41T^{2} \) |
| 43 | \( 1 - 8.60T + 43T^{2} \) |
| 47 | \( 1 + 2.85T + 47T^{2} \) |
| 53 | \( 1 + 2.24T + 53T^{2} \) |
| 59 | \( 1 - 0.271iT - 59T^{2} \) |
| 61 | \( 1 + 14.8iT - 61T^{2} \) |
| 67 | \( 1 + 3.34T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 10.4iT - 79T^{2} \) |
| 83 | \( 1 + 3.52iT - 83T^{2} \) |
| 89 | \( 1 - 0.708iT - 89T^{2} \) |
| 97 | \( 1 + 19.4T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.77241798583316529391764812077, −9.642186542932714332095937295781, −9.290244466120198408733161995476, −8.377178487831647697525293243399, −7.934358657802676046186492734208, −6.14605900604653547201721173815, −5.37415680992711829144718777188, −3.58649973586547202641280795106, −2.82977885307830635735823687360, −1.77382083660769930759276343586,
1.17426740881484636292113589879, 2.27749916217014478732451191508, 4.07217677155893567473495578190, 5.24712170093941355265085953152, 6.72264673234841804518217622706, 7.32964689345415319379322659545, 7.62223338320559755758200663579, 9.266764125459703335449867720617, 9.533981300085237123296832589384, 10.23891257149586245170466938020